One nice thing about "weak fibrations" (maps homotopy equivalent in the category of maps to Hurewicz fibrations) is that a pullback square involving (one) weak fibration is a homotopy pullback square.

Is the corresponding result
true for quasifibrations in the Serre-Quillen context? That is, suppose $E\to B$
is a quasifibration, and the square
$$
\begin{array}{ccc}
P & \to & E
\cr\downarrow&pb&\downarrow
\cr
X& \to &B
\end{array}
$$
is a categorical pullback. Then is it a homotopy pullback in the Quillen-Serre model
structure?

2 Answers
2

The definition of quasifibration (according to Dold & Thom, 1958) is: a map $f:E\to B$ such that for all $b$ in $B$, the canonical map from the fiber to the homotopy fiber is a weak equivalence. Pullbacks with respect to such maps are not generally homotopy pullbacks; an example was given in that 1958 paper (Bermerkung 2.3), which goes something like this:

Let $\newcommand{\R}{\mathbb{R}}B=\R\times \R$. Then $E$ will have the same underlying set as $B$, and $f$ will be the identity map. But we topologize $E$ by "tearing" along the positive $y$-axis. For instance, let $E$ have the smallest topology such that $f$ is continuous and the set $[0,\infty)\times (0,\infty)$ is open.

The space $E$ is still contractible with this topology (it deformation retracts to $\R\times -1$). Therefore, the homotopy fiber over any point of $B$ must be weakly contractible, and thus weakly equivalent to the actual one-point fiber. So $f$ is a quasi-fibration.

Let $X= \mathbb{R}\times 1\subset B$, and let $P$ be the pullback of $E$ over $X$. Then $P$ has two path components, while $X$ is contractible; this is not a homotopy pullback!